The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

Abstract

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct totally geodesic surfaces?